Space-ivision Relay: A High-Rate Cooperation Scheme for Fading Multiple-Access Channels Arumugam Kannan and John R. Barry School of ECE, Georgia Institute of Technology Atlanta, GA 0-050 USA, {aru, barry}@ece.gatech.edu Abstract We propose a new cooperation protocol for the fading multiple-access channel called space-division relay (SR). It is similar to the protocol of Laneman, Tse and Wornell (LTW) [], except that the relays use spacedivision instead of time-division multiplexing. The result is a nonorthogonal cooperation protocol with a higher rate that nevertheless achieves the full diversity of the two-user Rayleigh-fading cooperative multiple-access channel, a result that is proven in this paper. We examine the SR protocol with two types of relays: the amplify-andforward (AF) relay and the amplify/decode-and-forward (AF) relay. We derive the outage probability of the proposed scheme with AF relays. We present numerical results for the two-user channel at bps/hz which show that SR-AF outperforms all previously reported cooperative schemes. In particular, in terms of the SNR required to achieve a 0 outage probability, the SR-AF protocol outperforms the LTW-AF protocol by.9 db, while the SR-AF protocol outperforms another nonorthogonal protocol called NAF [4] by db, falling only. db short of the ideal cooperation bound. I. INTROUCTION This paper concerns the cooperative multiple-access channel, where two or more users wish to send independent messages to a common destination, and where these users cooperate by occasionally acting as relays for each other. The users adhere to a half-duplex constraint, preventing them from transmitting and receiving at the same time. By sharing their antennas and signal processing resources, the users together create a virtual transmit array []-[9] that provides each user with an additional diversity against fading, thereby increasing the reliability of communication. A noncooperative multiple-access strategy like SMA can achieve a high rate but with low diversity. In contrast, because user cooperation necessitates that the users spend some fraction of time listening to other users and acting as relays, a cooperative strategy will have a lower rate and a higher diversity []. There is a fundamental tradeoff in cooperative systems between rate and diversity. The technical This research was supported in part by National Science Foundation grants 040 and 0565, and by Texas Instruments. challenge is to design a protocol that achieves full cooperation diversity while keeping the rate loss to a minimum. Cooperative multiple-access protocols can be classified as either orthogonal or nonorthogonal. Orthogonal protocols, such as the LTW protocol [], are those in which different users are constrained to transmit in nonoverlapping time or frequency subchannels, thereby avoiding interference. These protocols have the advantage of simple decoding, but suffer from low rates due to the orthogonality constraint, and consequently result in high outage probabilities. Nonorthogonal protocols, such as the NAF protocol [4], allow simultaneous transmission among users. This enables a higher rate at the expense of higher decoding complexity. This paper focuses primarily on a three-node multipleaccess system, with two users sending independent information to a common destination. We propose a new nonorthogonal cooperation protocol called the space-division relay (SR) protocol. It modifies the LTW protocol [] to use space-division multiplexing instead of time-division multiplexing for the relays. We show that SR achieves the best outage performance among all previously reported orthogonal and nonorthogonal cooperation protocols for the two-user Rayleigh-fading multiple-access channel. This paper is organized as follows. In Section II, we describe the channel model and system assumptions. In Section III, we describe the SR protocol. In Section IV, we derive the outage probability and diversity order of the SR protocol. In Section V, we present some numerical results. In Section VI we present our conclusions. II. CHANNEL MOEL We consider a Rayleigh-fading multiple-access channel with two users communicating with a common destination. Each of the three nodes is equipped with a single antenna. To simplify our presentation we consider a completely symmetric scenario: () both users have an identical target spectral efficiency of R bps/hz; () both have an identical average energy of E per signalling interval; and () the average path loss between users is identical to the average path loss from each user to the destination. Asymmetry in any of these variables is easily incorporated into the system design without affecting the design principle.
Let h i denote the channel gain between the i-th user and the destination, and let h denote the channel gain between the two users. The channels are assumed to be linear and flat fading over the signal bandwidth. Also, the channels are assumed to be quasistatic, so that the channel response is constant over a frame consisting of T symbol periods, and it changes to an independent value from one frame to the next. The channel coefficients {h,h,h } are i.i.d. circularly symmetric complex Gaussian random variables with zero mean and unit variance. The additive noise at each receiving terminal is independent circularly symmetric Gaussian random variable with zero mean and variance N 0. Under these assumptions, the SNR of each user at any receiver is S = E/N 0. We assume that the users are frame-synchronized. We further assume that the destination knows all of channel coefficients {h,h,h }, whereas the users know only h. III. THE SPACE-IVISION PROTOCOL An illustration of the proposed SR protocol is shown in Fig.. The static fading frame is divided into three equalsized blocks of duration T/ signaling intervals. uring the first block, the first user (U ) transmits its information, while the second user (U ) and the destination () each listen to the transmission. uring the second block, U transmits its own independent information, while U and listen to the transmission. This completes the direct transmission part of the cooperation protocol for one frame. uring the third block, both users relay the received packets from the other user simultaneously, in a spacedivision multiple access fashion, so that the destination receives a linear combination of these two transmissions. Initially, we assume that the nodes use the amplify-andforward (AF) relaying technique []. The motivation of the SR protocol is to increase the rate compared to the LTW protocol by relaxing the orthogonality U U U U U U SPACE-IVISION U U constraint. Specifically, whereas the rate of each user in the LTW protocol is /4, the rate of each user in the SR protocol is /. Nevertheless we will see that SR still achieves full cooperative diversity. The cooperation scheme can be summarized as follows. uring the first block, the first user transmits {x (), x ( T/)} with average symbol energy E t = E[ x ( i ) ], while the second user listens. The received samples at and U are given by y ( i ) = h x ( i ) + n ( i ), y ( i ) = h x ( i ) + n ( i ), () for i {,, T/}. uring the second block, the first user listens, while the second user transmits its own information symbols {x (), x ( T/)}. The samples received by and U are given by y ( i ) = h x ( i ) + n ( i ), y ( i ) = h x ( i ) + n 4 ( i ), () for i {,, T/}. The third block is the relay phase. uring the third block in the case of AF relays, user transmits {αy (), αy ( T/)} while user simultaneously transmits {αy (), αy ( T/)}, where α is the amplification factor: α = -------------------------. () N h 0 + ------ E t The samples received by the destination during the third block are thus y ( i )=h αy ( i ) + h αy ( i ) + n 5 ( i ), (4) for i {,, T/}. Since each node is silent / of the time, the average power constraint is satisfied by choosing E t =E/. Instead of AF, the nodes could also use the amplify/ decode-and-forward (AF) relay technique [6]. An AF relay will use its knowledge of the channel coefficients to make a decision to either act as an AF relay or a decode-andforward relay. Specifically, if the interuser channel is not in outage, i.e., if log ( +.5 h E) > R, then each user can perfectly decode the other s information, and hence forward a clean version to the destination. The AF relay will thus act as a decode-and-forward relay. On the other hand, in the case of an outage, it would be counter-productive to forward erroneously decoded information, so the AF relay simply amplifies and forwards the samples instead. This hybrid relay strategy was shown to be better than both AF and F [6]. Fig.. Illustration of the space-division relay cooperation protocol.
In the following section, we derive an expression for the outage probability and the diversity order of the SR protocol with AF relays. IV. OUTAGE ANALYSIS In this section, we derive the outage probability of the SR cooperation protocol with AF relays. In the SR protocol, the observations at the destination consist of three received blocks Y, Y, and Y, where Y i = [y i (), y i () y i (T/)], corresponding to the two blocks X and X transmitted by the two users, where X i = [x i (), x i () x i (T/)]. The discrete, memoryless multiple-access channel created by the SR protocol is then: X Y = Y = H + AN, (5) Y X where the (i, j)-th element of the noise matrix N is n i ( j ), and where the matrices H and A given by h 0 0 0 0 0 H = 0 h, A = 0 0 0 0.(6) αh h αh h 0 0 αh αh Also, let H and H denote the first and second columns of H. The outage event for this multiple access system is the union of the following three events []: O : C = Y max T -- p( x ) I(X ; Y X ) < R O : C = max I(X ; Y X ) < R T -- p( x ) O : C = max ------ I(X, X ; Y ) < R, (7) p( x ) T where p( x ) is the joint probability density function of X and X. The outage probability of thus: P o = Pr[O O O ] = Pr[min{C, C, C } < R]. (8) These quantities can be shown to be maximized when the input alphabet at each source follows an independent Gaussian distribution. Upon maximization, we get C = -- log det(i + -- SH * (AA * ) H ) C = -- log det(i + -- SH * (AA * ) H ) C = -- log det(i + -- SH * (AA * ) H). (9) 6 Intuitively, the factor / represents the fact that the sources transmit new information only / of the total time. The expression for C further simplifies to: C = -- log det( + -- SH * (AA * ) H ) (0) S α h = -- log --S h h +,() + ---------------------------------------- Δ' where Δ' α = + ( h + h ). Substituting for α and simplifying, we get C = -- log S h --S h h +, () + ------------------------------- Δ where Δ = ( + S( h + h + h )). The expression for C can be obtained by exchanging h and h in the above expression, and C can be evaluated by similar substitutions. Thus, the outage probability of the SR protocol can be evaluated by substituting (9) and () into (8). We now briefly discuss the rate and diversity aspects of the SR scheme. efinition. The rate μ of a cooperative multiple-access protocol is the average number of information symbols transmitted by each user per signalling interval. For a two-user CMA system, TMA has rate /, while SMA has rate. The LTW protocol has rate /4, whereas the SR protocol has rate /. efinition. The diversity order d of a cooperative system is defined as logp d = o ( SR, ) lim ---------------------------------, () S logs where P o (S, R) is the outage probability of the scheme. For a two-user CMA system with one antenna at each node, TMA and SMA achieve a diversity order of just one, whereas the LTW protocol achieves the full diversity (d = ). We next show that the SR protocol with AF achieves full diversity. The diversity order of the SR protocol can be computed as follows. Let P =Pr[O ], P = Pr[O ], and P = Pr[O ]. Let the diversity order corresponding to P, P and P be d, d and d, respectively. Note that the assumption that h and h are statistically identical implies that d = d. The outage probability can be bounded using the union bound as P δ P o P + P + P, (4) where P δ is either of P, P or P. We state the following theorem on the diversity order of SR.
Theorem. The SR protocol for a two-user cooperative multiple-access channel with one antenna at each node achieves the full diversity order of d =. Proof: (Sketch) Using the bound in (4), it is easy to show that d SR = min{d, d, d }. Using the inequality I(X, X ; Y ) I(X ; Y X ), we see that C C, implying that d d. Using (), the probability P is given by S h P = Pr h h ( R ) + ------------------------------- < -------------------------. (5) Δ S Using the fact that h, h and h are i.i.d. exponentially distributed random variables, and employing transformation of random variables, it can be shown that P decays as S for large values of S, thus completing the proof. The outage probability of SR with AF can be derived in a similar fashion and can be shown to be strictly less than SR with AF. Consequently, SR with AF also achieves full diversity. In terms of the diversity-multiplexing framework of Zheng and Tse [0], the diversity-multiplexing tradeoff of each user in the two-user SR protocol can be shown to be d( ρ ) = ( ρ), for 0 ρ /, (6) where the multiplexing gain ρ of each user is at most /. V. NUMERICAL RESULTS In this section, we present numerical results for a Rayleigh-fading cooperative multiple-access system with two users and a single destination, each equipped with one antenna. Each user has a target spectral efficiency of R =bps/hz, and each has the same average SNR. To achieve this target spectral efficiency, the LTW protocol needs a user to transmit information at 4 bps/hz during its active transmissions, while SR and NAF require the user to transmit at bps/hz and bps/hz respectively when active. In Fig., we compare several candidate schemes by plotting the outage probability versus SNR. Traditional multiple access schemes such as TMA and SMA perform well at low SNR, but their performance suffers from a lack of diversity at high SNR. In contrast, the benefits of cooperative diversity (LTW and SR) at high SNR are clearly evident. At an outage probability of 0, SR with AF outperforms LTW with AF by.9 db. A similar result (not shown) is observed with AF relays as well. Also shown in the figure (labeled co-located bound) is the outage probability of a MISO channel, which serves as a lower bound on the outage OUTAGE PROBABILITY 0-0 - 0-0 -4 0-5 CO-LOCATE BOUN ( MISO) 0-6 0 0 0 0 SNR (db) SR - AF TMA SMA LTW - AF Fig.. Comparison of outage probabilities of various multiple access schemes for a -user system, with R = bps/hz. probability of any CMA scheme, although it may not be achievable. We see from Fig. that SR with AF falls 4.7 db short of the MISO bound. In Fig., we compare the performance of three nonorthogonal cooperation protocols: SR with AF, SR with AF, and the NAF protocol. We see that SR with AF is.4 db worse than NAF. However, SR with AF outperforms NAF by db. Therefore, SR-AF achieves the best outage performance among all previously reported cooperation protocols, falling only. db short of the MISO bound. For R = bps/hz, SR with AF outperforms NAF by. db and LTW with AF by 4.5 db at an outage probability of 0. OUTAGE PROBABILITY 0-0 - 0-0 -4 0-5 SR - AF CO-LOCATE BOUN ( MISO) Fig.. Comparison of outage probabilities of non-orthogonal cooperation schemes for a -user system, with R = bps/hz. NAF SR - AF 0-6 0 0 0 0 SNR (db)
Both SR and NAF achieve full diversity and the rate of NAF (/) is higher than that of SR (/). However, the NAF protocol suffers a power penalty because of the need for each user to share its energy between current and past symbols of its own and the other user, a drawback not captured by the definition of rate. Moreover, the sequential nature of NAF makes it incompatible with the AF strategy. This explains the inferior performance of NAF when compared to SR-AF, despite its higher rate. We note that the relative performance of these multipleaccess strategies depends strongly on the target spectral efficiency and SNR. Since the rate of SR is higher than that of LTW, the SNR improvement over LTW increases as the target spectral efficiency increases. For the same reason, NAF outperforms SR at a sufficiently high spectral efficiency, beyond about 4 bps/hz. However, for R >.5 bps/hz, it turns out that SMA requires even less SNR. Overall, of the multiple-access strategies compared in this paper, the best outage performance can be achieved by switching between SR and SMA as the spectral efficiency and SNR vary. Though nonorthogonal protocols outperform orthogonal protocols, they typically have a higher decoding complexity. Roughly, in terms of the alphabet size M, the decoding complexity of LTW scales as O( M ), whereas the decoding complexity of SR and NAF scales as O( M ). The SR protocol is even more advantageous when there are more than users. SR can be extended to a multipleaccess system with N users by employing space-division multiplexing over the transmission phase as well as the relay phase. For example, in Fig. 4 we illustrate how the SR protocol applies to the case of N = users. From the figure we see that the rate of each user is /. In general, the rate of the SR protocol for an N-user system is (N )/(N +). Interestingly, the rate of the SR protocol per user grows with N. In stark contrast, the rate for LTW is /N, while the rate for NAF is /N. espite the high rate, the SR protocol is sufficient to ensure good diversity performance. For example, on the -user Rayleigh-fading multiple access channel, SR-AF outperforms the corresponding LTW-AF protocol by 7. db. U U VI. CONCLUSIONS We proposed a new cooperative multiple-access strategy called space-division relay (SR). We introduced spacedivision relay as a simple nonorthogonal cooperation protocol that achieves the full cooperative diversity. SR uses spacedivision multiplexing during its relay phase to achieve a higher transmission rate. We investigated SR with both amplify-and-forward and amplify/decode-and-forward relays. We showed that the high rate of SR-AF enables it to outperform the LTW-AF protocol by.9 db at an outage probability of 0 at a target spectral efficiency of bps/hz. We also showed that SR-AF outperforms NAF by db. We also observe that SR-AF achieves the best outage performance among all previously reported protocols, falling only. db short of the ideal cooperation bound. VII.REFERENCES [] J. N. Laneman,. N. C. Tse and G. W. Wornell, Cooperative iversity in Wireless Networks: Efficient Protocols and Outage Behavior, IEEE T. Info. Theory, 50:, pp 06 080, ec. 004. [] A. Sendonaris, E. Erkip and B.Aazhang, User Cooperation iversity Part I: System escription, IEEE Transactions on Communications, V. 5, No., pp. 97 98, Nov. 00. [] J. N. Laneman, Cooperative iversity in Wireless Networks: Algorithms and Architectures, Ph.. issertation, MIT, MA, 00. [4] K. Azarian, H. El Gamal, P. Schniter, On the achievable diversitymultiplexing tradeoff in half-duplex cooperative channels, IEEE Trans. on Info. Theory, Vol. 5, No., pp 45-47, ec. 005. [5] K. Azarian, H. El Gamal and P. Schniter, On the esign of Cooperative Transmission Schemes Conference, Allerton Conf. on Commun, Control and Computing, Monticello, IL, 00. [6] X. Bao and J. Li, ecode-amplify-forward A New Class of Forwarding Strategy for Wireless Relay Channels, Proc. IEEE SPAWC, New York, pp. 86-80, June 005. [7]. Gesbert, A. Hjørungnes, H. Skjevling, Cooperative Spatial Multiplexing with Hybrid Channel Knowledge, Proc. International Zurich Seminar on Communications, Feb. 006. [8] L. Lai, K. Liu, and H. El Gamal, The Three Node Wireless Network: Achievable Rates and Cooperation Strategies, IEEE Transactions on Information Theory, Vol. 5, No., pp. 805-88, March 006. [9] N. Prasad and M. K. Varanasi, iversity and Multiplexing Tradeoff Bounds for Cooperative iversity Protocols, Proc. IEEE Intl. Symposium on Information Theory, Chicago, IL, p. 68, June 004. [0] avid N. C. Tse, Pramod Viswanath and Lizhong Zheng, iversitymultiplexing tradeoff in multiple-access channels, IEEE Trans. Information Theory, Vol. 50, No. 9, pp. 859-874, Sept. 004. [] T. M. Cover and J. A. Thomas, Elements of Information Theory, John Wiley and Sons, 99. U Fig. 4. Extension of the SR protocol to N = users.